| L(s) = 1 | + (−0.336 − 1.37i)2-s + (2.62 − 1.01i)3-s + (−1.77 + 0.925i)4-s + (1.14 − 1.46i)5-s + (−2.27 − 3.26i)6-s + (0.147 − 0.410i)7-s + (1.86 + 2.12i)8-s + (3.66 − 3.31i)9-s + (−2.39 − 1.07i)10-s + (−0.0793 − 0.353i)11-s + (−3.72 + 4.23i)12-s + (0.384 − 0.106i)13-s + (−0.613 − 0.0635i)14-s + (1.51 − 5.00i)15-s + (2.28 − 3.28i)16-s + (−0.207 + 0.0628i)17-s + ⋯ |
| L(s) = 1 | + (−0.238 − 0.971i)2-s + (1.51 − 0.585i)3-s + (−0.886 + 0.462i)4-s + (0.510 − 0.654i)5-s + (−0.930 − 1.33i)6-s + (0.0555 − 0.155i)7-s + (0.660 + 0.750i)8-s + (1.22 − 1.10i)9-s + (−0.757 − 0.340i)10-s + (−0.0239 − 0.106i)11-s + (−1.07 + 1.22i)12-s + (0.106 − 0.0294i)13-s + (−0.164 − 0.0169i)14-s + (0.392 − 1.29i)15-s + (0.572 − 0.820i)16-s + (−0.0502 + 0.0152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05310 - 1.77315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05310 - 1.77315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.336 + 1.37i)T \) |
| good | 3 | \( 1 + (-2.62 + 1.01i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (-1.14 + 1.46i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (-0.147 + 0.410i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (0.0793 + 0.353i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (-0.384 + 0.106i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (0.207 - 0.0628i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (-1.28 - 2.55i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (6.53 - 0.969i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (0.762 + 4.39i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (-3.26 + 4.89i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (7.11 - 6.13i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (-0.336 + 1.34i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (-2.57 - 5.81i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (-4.68 - 5.70i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (-0.818 + 4.71i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (-0.910 + 3.28i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (3.22 + 0.0792i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (9.38 - 8.93i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (5.39 - 0.264i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (3.29 + 9.20i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (-0.0361 + 0.00356i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (-5.21 - 4.49i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.77 - 0.856i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (-8.07 + 1.60i)T + (89.6 - 37.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31098855745823733862731410880, −9.593347239886929494210691409135, −8.921802902777397187478821916995, −8.124029750509771908785571706635, −7.53148498508753889910232586795, −5.89384181613339189745467944348, −4.42561420284266483735433402004, −3.41862759543584591827066899225, −2.29139371009158856074686813547, −1.33958788234710921893616512480,
2.14806800601370073087744924130, 3.42437804280357550651142706196, 4.48184296378715569263528781192, 5.69869788460029417324580049299, 6.86959338025146475588113361440, 7.64738893760536566311112536236, 8.711756787122355029224994903290, 9.051521476829480407158737237542, 10.19557906365174743130869761118, 10.46378509499735746406424736156
